Elementary Linear Algebra, 11th Edition

CHAPTER 8General Linear Transformations

CHAPTER CONTENTS

1. 8.1 General Linear Transformations
2. 8.2 Compositions and Inverse Transformations
3. 8.3 Isomorphism
4. 8.4 Matrices for General Linear Transformations
5. 8.5 Similarity

INTRODUCTION

In earlier sections we studied linear transformations from $R^n$$R^m$$V$$W$

8.1 General Linear Transformations

Up to now our study of linear transformations has focused on transformations from $R^n$$R^m$$n$$R^n$

Definitions and Terminology

In Section 1.8 we defined a matrix transformation $T_A:R^n\to R^m$

$$T_A\left(\mathbf{x}\right)=A\mathbf{x}$$

in which $A$$m\times n$Theorem 1.8.3 that the matrix transformations are precisely the linear transformations from $R^n$$R^m$

$$\begin{array}{ccc}\hfill T\left(\mathbf{u}+\mathbf{v}\right)=T\left(\mathbf{u}\right)+T\left(\mathbf{v}\right)\hfill & \hfill \text{and}\hfill & \hfill T\left(k\mathbf{u}\right)=kT\left(\mathbf{u}\right)\hfill \end{array}$$

We will use these two properties as the starting point for defining more general linear transformations.

DEFINITION 1

If $T:V\to W$$V$$W$$T$linear transformation from $V$$W$$\mathbf{u}$$\mathbf{v}$$V$$k$

1. $\begin{array}{cc}T\left(k\mathbf{u}\right)=kT\left(\mathbf{u}\right)\hfill & \left[\mathbf{\text{Homogeneity property}}\right]\hfill \end{array}$
2. $\begin{array}{cc}T\left(\mathbf{u}+\mathbf{v}\right)=T\left(\mathbf{u}\right)+T\left(\mathbf{v}\right)\hfill & \left[\mathbf{\text{Additivity property}}\right]\hfill \end{array}$

In the special case where $V=W$$T$linear operator on the vector space $V$

The homogeneity and additivity properties of a linear transformation $T:V\to W$${\mathbf{v}}_1$${\mathbf{v}}_2$$V$$k_1$$k_2$

$$T\left(k_1{\mathbf{v}}_1+k_2{\mathbf{v}}_2\right)=k_{1}T\left({\mathbf{v}}_1\right)+k_{2}T\left({\mathbf{v}}_2\right)$$

More generally, if ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_r$$V$$k_1,k_2,\dots ,k_r$

(1)$$\boxed{T\left(k_1{\mathbf{v}}_1+k_2{\mathbf{v}}_2+\cdots +k_r{\mathbf{v}}_r\right)=k_{1}T\left({\mathbf{v}}_1\right)+k_{2}T\left({\mathbf{v}}_2\right)+\cdots +k_{r}T\left({\mathbf{v}}_r\right)}$$

The following theorem is an analog of parts (a) and (b) of Theorem 1.8.2.

THEOREM 8.1.1

If $T:V\to W$

1. $T\left(\mathbf{0}\right)=\mathbf{0}$
2. $T\left(\mathbf{u}-\mathbf{v}\right)=T\left(\mathbf{u}\right)-T\left(\mathbf{v}\right)$$\mathbf{u}$$\mathbf{v}$$V$

Use the two parts of Theorem 8.1.1 to prove that

$$T\left(-\mathbf{v}\right)=-T\left(\mathbf{v}\right)$$

for all $\mathbf{v}$$V$

Proof

Let $\mathbf{u}$$V$$0\mathbf{u}=\mathbf{0}$Definition 1 that

$$T\left(\mathbf{0}\right)=T\left(0\mathbf{u}\right)=0T\left(\mathbf{u}\right)=\mathbf{0}$$

which proves (a).

We can prove ...